Describe a point's motion on a chosen straight axis, then connect signed changes, instantaneous rates and one exact set of position, velocity and acceleration graphs.
Key knowledge
In a chosen reference frame, instantaneous velocity is the time derivative of position and instantaneous acceleration is the time derivative of velocity.
Values and equations
v = dr/dt; a = dv/dt
r
Position vector in the chosen frame m
v
Velocity vector m s⁻¹
a
Acceleration vector m s⁻²
t
Time s
Conditions and scope
Velocity requires the position derivative; acceleration also requires the relevant derivative of velocity, or second derivative of position, to exist.
The relations describe motion; they do not alone specify its cause.
Sources · 6
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
OpenStax 3.3, equation 3.9 and its relation to the derivative of position.
2026-10-10 · Classical kinematic definitions only; units and derivative requirements checked.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Scope or convention differs
In one-dimensional motion, velocity and acceleration are successive time derivatives of position.
The chapter verifies the one-dimensional case; the repository also states the spatial-vector form and frame requirement.
National Council of Educational Research and Training (NCERT)
Displacement is the final position minus the initial position in a stated frame. Distance is the length of the travelled path and is at least the displacement magnitude. Average velocity is displacement divided by elapsed time; average speed is distance divided by that time. A round trip can have zero average velocity and positive average speed.
This relation concerns the stated displacement, distance and average speed claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
For constant vector acceleration a, position and velocity evolve as v=v₀+aΔt and r=r₀+v₀Δt+aΔt²/2. In one dimension, v²=v₀²+2a(x−x₀). The formulas describe free fall only when gravity is adequately uniform and other forces are negligible.
This relation concerns the stated constant-acceleration kinematics claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
For a particle on a circle of fixed radius R, acceleration has inward radial component v²/R and tangential component dv/dt. Uniform circular motion has v=Rω and inward acceleration Rω². “Centripetal force” names the inward net-force component required by the motion; tension, gravity or another interaction supplies it.
This relation concerns the stated circular motion and centripetal acceleration claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
Average power is work divided by the time interval; instantaneous mechanical power is dW/dt. For a force acting at a point moving with velocity v, its power is F·v. A force perpendicular to that velocity transfers no mechanical energy through that point even if it changes the direction of motion.
This relation concerns the stated mechanical power claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
Undamped one-dimensional motion obeying m ẍ=−kx has x=Acos(ωt+φ), with ω=√(k/m), period T=2π/ω and frequency f=1/T. Acceleration is −ω²x. Initial position and velocity determine amplitude and phase; frequency is independent of amplitude only within the ideal linear model.
This relation concerns the stated simple harmonic motion and period claim and its conditions; it does not assert an unqualified prerequisite or an additional theorem.
Learning guidance · 4 / 4
Preserved learning suggestions. They are not proof, empirical evidence or mandatory prerequisites.